Looking at the ifixit Unsticking SteelSeries Nimbus Triggers guide, it shows the battery which has the following specification FT712257P 3.7V 820mAH.
Using this information I believe that the dimensions of the battery are 7.1mm depth x 22mm width x 57mm length.
If these dimensions are correct then it seems as though the battery is a special size just made for the manufacturer of the controller.
I found this battery that may be a suitable replacement, given that the measurements above are correct. It is rechargeable, has the correct voltage, nearly the same capacity (mAH) and most importantly it should physically fit into the controller.
There are some drawbacks if you decide to try installing this battery as a replacement.
It may have to be soldered into place rather than plugged in as it only has a two pin connector which most probably is not suitable for the 3 pin connector on the motherboard. Perhaps you could cut off the new battery 2 pin plug and cut the wires off the old battery at the battery end and then solder the red wire to the red wire and the black wire to the black wire , joints insulated with heatshrink tubing, so that the new battery is using the old battery plug connector. Just a thought
It hasn’t got a 3rd wire for the battery management control (usually tied to a thermistor attached to the battery case inside the packaging) so it may not charge when connected depending on the charging circuit monitoring arrangement that is used in the controller. If it does charge then the charging will have to be supervised as there will be no cut-off arrangement if the battery gets too hot after it is fully charged and the charger has not been disconnected, which can create problems, even a fire. Perhaps an audible alarm timer could be set as a reminder to disconnect the charger after a suitable charging time has elapsed
The battery compartment may have to be padded out a bit to hold the battery in place as it is a bit shorter than the original
Given the cost of the battery (didn’t check what the shipping costs are or how long it would take to deliver) it may be an option to consider to keep the controller working rather than being discarded.